In Wikipidia it's said that the Tanaka's formula is the explicit Doob–Meyer decomposition of the submartingale $\vert B(t)\vert$.
In general we have that Tanaka's formula could be written as:
$$\vert B(t)-a\vert =\vert a\vert+ \int_0^t {\text{sgn}}(B(s)-a)dB(s)+\mathbb P\lim_{\epsilon\to0}\frac{1}{2\epsilon}\int_0^t1_{(a-\epsilon,a+\epsilon)}(B(s))ds$$
I can see that since $\vert{\text{sgn}}(B(s)-a)\vert \leq 1$ we have that $$\mathbb E\int_0^t \vert{\text{sgn}}(B(s)-a)\vert^2ds<\infty,$$
and hence $M_t=\int_0^t {\text{sgn}}(B(s)-a)dB(s)$ is a continuous martingale.
Then I should be able to show that $\mathbb P\lim_{\epsilon\to0}\frac{1}{2\epsilon}\int_0^t1_{(a-\epsilon,a+\epsilon)}(B(s))ds$ (the local time) is a predictable, rigt continuous increasing processes.
This may be straightforward but I am not able to see the latter. Could you please give me some hint? Thanks in advance.